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198       XIII    INTEGRATION

The last assertion follows from the definition of equivalent measures. To
prove the first assertion, we may, by virtue of (13.15.7), assume that the
measures un are bounded. By multiplying each jun by a suitable positive real
number, we may also assume that jun(l) :g 2~M.

If  vn = sup(jKl9//2, ... , /O>   then   by   virtue   of   (13.15.3.2)   we   have

vn(f) g /4(/) g 1 for every /e JfR(X) such that 0 g/^ 1, and every n.

k-l

Hence (13.4.4) the increasing sequence (vn) converges vaguely to its least upper

bound v in MR(X), and we have v(/) :g 1 for every /e Jf R(X) such that
0 ^/^ 1. Hence v*(l) g 1 and so v is bounded. We may therefore write
l*n 9n' v> where # is locally v-integrable and ^0; hence vn  i sup gk\  v

\l**n     /

(13.15.3.3) and consequently v=(sup#w\-v  (13.15.3.5).  It is now clear

\ n        /

(1 3.15.5) that the relation v(N) = 0 implies that nn(N) = 0 for all n ; conversely,
if /zn(N) = 0for allw, then each of the functions gn <pN is v-negligible (13.14.1),
hence so is /sup gn\q>N by (13.6.2), and so N is v-negligible.

(13.15.9) Let (un) be an increasing sequence of positive measures having a
least upper bound v in MR(X). For a function/ to be v-integrable it is necessary
and Sufficient that f should be nn~integrable for all , and that

sup I \f\dpn< +00.

n

In that case we have

Ifdv = lim   /d/jn.

J                 n-*o> J

We may write fin  gn  v, where gn is ^0 and locally v-integrable, and the
sequence (gn) is increasing. Moreover, we have v =/sup gn\  v by (13.15.3),

\   n         J

hence sup gn = 1 almost everywhere with respect to v. If/is v-integrable, then

n

so is/0,, for all /j, by virtue of (13.9.6), the inequality \fgn\ <; |/| (which is true
almost everywhere with respect to v), and (13.9.13). Hence by (13.14.3) /is
/V-integrable for all n. Conversely, if/is ^n-integrable for all n, then the func-
tions^, are v-integrable (13.14.3) and we have/= lim^n almost every-

-oo

where with respect to v, so that / is v-measurable (13.9.11). Also we have
I/I = sup \fgn\ almost everywhere with respect to v. Hence the result follows

n

from (13.9.13) and (13.8.1), and the last assertion of the proposition follows
from the dominated convergence theorem (13.8.4). v). Furthermore, if v' is another positive measure on X